My federal return was rejected because a dependent child was claimed on another return for EIC as well as mine. I had filed the return electronically. My NJ state return was filed by mail and I haven't heard anything yet about it being rejected. Now that I've corrected my federal return I owe the state more money. Do I need to amend my state return or should I expect that my state return will also be rejected?

If the NJ return was filed by mail, then the rejection has not yet filtered down to the state.

Amend the state return as soon as possible. If you come forward and fix the problem, NJ will probably favorably consider any request to waive penalties and possibly interest. They will NOT be so understanding if they have to come to you for the extra money.